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 A206496 The Connes-Moscovici weight of the rooted tree with Matula-Goebel number n. It is defined as the number of ways to build up the rooted tree from the one-vertex tree by adding successively edges to the existing vertices. 2
 1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 3, 4, 10, 1, 1, 15, 1, 10, 10, 5, 3, 10, 10, 15, 15, 10, 4, 60, 1, 1, 15, 5, 20, 45, 6, 5, 45, 20, 3, 60, 4, 15, 105, 18, 10, 15, 10, 70, 15, 45, 1, 105, 35, 20, 15, 24, 1, 210, 15, 6, 105, 1, 105, 105, 1, 15, 63, 140 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS See A206494 for the number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. REFERENCES F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143. I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142. I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22. D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273. J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations, 1987), Wiley, Chichester. Ch. Brouder, Runge-Kutta methods and renormalization, Eur. Phys. J. C 12, 2000, 521-534. D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, J. Symbolic Computation, 27, 1999, 581-600. J. Fulman, Mixing time for a random walk on rooted trees, The Electronic J. of Combinatorics, 16, 2009, R139. LINKS E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288. FORMULA a(n) = V(n)!/[TF(n)*SF(n)], where V denotes "number of vertices" (A061775), TF denotes "tree factorial" (A206493), and SF denotes "symmetry factor" (A206497). EXAMPLE a(6)=3 because the rooted tree with Matula-Goebel number 6 is the path ARBC with root at R; starting with R we can obtain the tree ARBC by adding successively edges at the vertices  (i) R, R, A or at (ii) R, R, B, or at (iii) R, A, R. a(8)=1 because the rooted tree with Matula-Goebel number 8 is the star tree with 3 edges emanating from the root; obviously, there is only 1 way to build up this tree from the root. MAPLE with(numtheory): V := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: TF := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow; n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then V(n)*TF(pi(n)) else TF(r(n))*TF(s(n))*V(n)/(V(r(n))*V(s(n))) end if end proc: SF := proc (n) if n = 1 then 1 elif nops(factorset(n)) = 1 then factorial(log[factorset(n)[1]](n))*SF(pi(factorset(n)[1]))^log[factorset(n)[1]](n) else SF(expand(op(1, ifactor(n))))*SF(expand(n/op(1, ifactor(n)))) end if end proc: a := proc (n) options operator, arrow: factorial(V(n))/(TF(n)*SF(n)) end proc: seq(a(n), n = 1 .. 120); CROSSREFS Cf. A061775, A206493, A206497, A206494 Sequence in context: A117184 A035690 A124794 * A097560 A218905 A027960 Adjacent sequences:  A206493 A206494 A206495 * A206497 A206498 A206499 KEYWORD nonn AUTHOR Emeric Deutsch, Jul 20 2012 STATUS approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)